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# Fundamentals of gas for gas lift design

Only the gas fundamentals essential to the design and analysis of gas lift installations and operations are discussed in this section. The more important gas calculations related to gas lift wells and systems can be divided into these topics:

• Gas pressure at depth
• Temperature effect on the confined nitrogen-charged bellows pressure
• Volumetric gas throughput of a choke or gas lift valve port
• Gas volume stored within a conduit

The fundamental gas equations are based on:

• Pressure in pounds per square inch absolute (psia)
• Temperature in degrees Rankine (°R)
• Volume or capacity in cubic feet (ft3)

An exception is pressure difference in pounds per square inch (psi), which may be a difference in gauge or absolute units because the calculated pressure difference is the same. Generally, field measurements of pressure are in gauge readings; therefore, the volumetric gas throughput and gas-pressure-at-depth charts are in units of psig. The gas lift valve equations and calculations for bellows-charge and operating pressures in this page use gauge pressure.

## Gas pressure at depth

Prediction of injection-gas pressure at depth is essential for proper gas lift installation design and for analyzing or troubleshooting gas lift operations. Most gas-pressure-at-depth calculations are based on a static gas column. Pressure loss, because of friction from the flow of injection gas through a typical casing/tubing annulus, is negligible. The gas velocity in the annulus is considered negligible because the cross-sectional area of the annulus is so much larger than the port area of a gas lift valve. The maximum gas flow rate is limited by the valve port size. Only in annular flow, where the flow areas are reversed and large volumes of gas may be injected down a small tubing string, does pressure loss because of velocity become a concern. Eq. 1 is used for predicting the static bottomhole injection-gas pressures. ....................(1)

where

 Pio = injection-gas pressure at surface, psia, PioD = injection-gas pressure at depth, psia, e = Napierian logarithm base = 2.718..., γg = gas specific gravity (air = 1.0), dimensionless, D = true vertical depth of gas column, ft, = average gas-column temperature, °R, and = compressibility factor based on gas-column average pressure and temperature , dimensionless.

The depth used in the equation is the true vertical depth of the gas column. Because the gas compressibility factor is a function of the average pressure and temperature, the solution to this equation requires several iterations. Generally, the average pressure and temperature are assumed to be the arithmetic mean of the wellhead and bottomhole values. This assumption is reasonable because the increase in well temperature with depth tends to result in a relatively constant gas density with depth. A straight-line traverse will approximate an actual static injection-gas pressure-at-depth traverse and is used for the design of most gas lift installations.

## Temperature effect on the confined nitrogen-charged bellows pressure

There are many more bellows-charged than spring-loaded gas lift valves in service. Most of the bellows-charged valves have nitrogen gas in the dome and bellows. Because it is impractical to set each gas lift valve at its operating well temperature, the test-rack opening or closing pressure is set at a standard base temperature. Most manufacturers set their bellows-charged gas lift valves with the nitrogen-gas charge in the bellows at 60°F. Nitrogen was selected as the charge gas because:

• The compressibility factors for nitrogen at various pressures and temperatures are known
• Nitrogen is noncorrosive and safe to handle
• Nitrogen is readily available throughout the world
• Nitrogen is inexpensive

The temperature correction factors for nitrogen can be obtained from tables such as the one shown in Table 1.  Table 1 is calculated for a specific condition of temperature and pressure (nitrogen-charged bellows pressure of 1,000 psig at 60°F) and is based on the work of Winkler and Eads.  An equation for calculating the temperature correction factor, CT, at other conditions of temperature and pressure is shown at the bottom of the table. However, for most gas lift designs, unless pressures are considerably higher than 1,000 psig, Table 1 gives sufficient accuracy. CT is used to calculate the nitrogen-charged bellows pressure at 60°F for a given valve operating or unloading temperature at valve depth in a well. ....................(2)

where

 CT = temperature correction factor for nitrogen from PbvD at TvuD to Pb at 60°F, dimensionless, Pb = nitrogen-charged bellows pressure at 60°F, psig and PbvD = nitrogen-charged bellows pressure at valve temperature, psig.

If a more accurate calculation of CT is required, the alternative solution shown in Example Problem 1b may be used.

## An alternative solution for calculating nitrogen-charged bellows pressure at 60oF

If the CT from Table 1 is used to calculate the nitrogen-charged bellows pressure at the test-rack valve setting temperature for gas lift valves in a high-injection-gas-pressure system, the possible error in the test-rack opening pressures may prevent successful gas lift operations. If the operating injection-gas-line pressure exceeds a range of 1,200 to 1,500 psig, the following correlation, based on the work of Winkler and Eads, is recommended for calculating the gas lift valve nitrogen-charged bellows pressure in psig at the setting test-rack opening temperature of 60°F. ....................(3)

where

 P = Pb + Patm and T = TvD - 60

If Pb is less than 1,250 psia:

 A = 3.054E – 07 ( T ), B = 1 + 0.001934(T) and C = – 0.00226 (T – P).

If Pb is greater than 1,250 psia:

 A = 1.84E – 07 (T), B = 1 + 0.002298 (T) and C = –0.267 (T – P).

When Eq. 3 is used to calculate Pb, Eq. 4 is used to calculate CT: ....................(4)

Example Problem 1a A 1.5-in.-OD gas lift valve with a 1/4-in.-ID port (Ap/Ab = 0.064 from Table 2), nitrogen-charged bellows pressure at well temperature PbvD = 800 psig at 142°F. Calculate Pvo using Table 1 and Eqs. 2 and 5:

1. Determine CT from Table 1: CT = 0.845 for TvD = 142°F.
2. Using Eq. 2, solve for Pb: Pb = 0.845(800) = 676 psig at 60°F.
3. Using Eq. 5, calculate the test-rack opening pressure, Pvo:

When Eq. 3 is used to calculate Pb: P = 814.7, T = 82, A = 2.50428E – 05, B = 1.158588, C = –814.8853, and Pb = 678.3 psig at 60°F. Using Eq. 5 to calculate Pvo and Eq. 4 to calculate CT:

The difference between using Eq. 3 or Table 1 for calculating Pvo is only 3 psi.

Example Problem 1b A 1.5-in.-OD gas lift valve with a 1/4-in. ID port ( Ap / Ab = 0.064 from Table 2 ), nitrogen-charged bellows pressure at well temperature PbvD = 2,228 psig at 200°F. Calculate Pvo using Table 1 and Eqs. 2 and 5 :

1. Determine CT from Table 1 for TvD = 200°F: CT = 0.761.
2. Using Eq. 3 , solve for Pb : Pb = 0.761(2,228) = 1,695.5 psig at 60°F.
3. Using Eq. 5 , calculate the test-rack opening pressure, Pvo:

Using Eq. 3 to calculate PbvD : P = 2,242.7 and T = 140.

A = 2.576E – 05, B = 1.32172, C = –2,280.1, and Pb = 1,656 psig at 60°F.

Using Eq. 5 to calculate Pvo and Eq. 4 to calculate CT :

For the high-injection-gas-pressure system, note that the calculated test-rack opening pressure is higher using the CT from Table 1 to correct the nitrogen-charged bellows pressure from valve temperature in the well to the setting temperature of 60°F. The above data represent an actual 1,800-psig injection-gas system for gas lifting deep wells in Alaska. The operator had difficulty unloading and gas lifting these wells because the set test-rack opening pressures of the gas lift valves were too high.

## Volumetric gas throughput of an orifice or choke

The volumetric gas throughput of an orifice or choke is calculated on the basis of an equation for flow through a converging nozzle. This equation is complex and lengthy for noncritical flow. For this reason, gas passage charts are widely used for estimating the volumetric gas flow rate. A widely used equation for calculating the gas flow rate through an orifice, choke, or full-open valve port was published by Thornhill-Craver. ....................(6)

where

 qgsc = gas-flow rate at standard conditions (14.7 psia and 60°F), Mscf/D, Cd = discharge coefficient (determined experimentally), dimensionless, A = area of orifice or choke open to gas flow, in.2, P1 = gas pressure upstream of an orifice or choke, psia, P2 = gas pressure downstream of an orifice or choke, psia, g = acceleration because of gravity, ft/sec2, k = ratio of specific heats (Cp/Cv), dimensionless, T1 = upstream gas temperature, °R, Fdu = pressure ratio, P2/P1, consistent absolute units,

and critical-flow pressure ratio, dimensionless.

If FduFcf, then Fdu = Fcf (critical flow). The gas-compressibility factor is not included in Eq. 6; therefore, most published gas passage charts do not include a gas-compressibility factor correction. Since the compressibility factor would enter the equation as a square root term in the denominator, the chart values will be lower than actual values for most injection-gas gravities and pressures. One type of choke capacity chart is illustrated in Figs. 1 and 2. The advantages of this type of display are the number of orifice sizes on a single chart for a full range of upstream and downstream pressures and that an orifice size can be determined for a given gas rate throughput and the given upstream and downstream pressures. The gas throughput capacity of the different orifice sizes is based standard conditions of 14.65 psia and 60°F for a gas gravity of 0.65 and an orifice discharge coefficient of 0.865.

Because gas flow in a gas lift installation occurs at the gas temperature at valve depth, a correction for temperature improves the prediction for the volumetric gas rate. If the actual gravity differs from 0.65, a second correction should be applied. An approximate correction for gas passage can be calculated using Eq.7. ....................(7)

and ....................(8)

where

 CgT = approximate gas gravity and temperature correction factor for choke charts, dimensionless, TgD = gas temperature at valve depth, °R, qga = actual volumetric gas rate, Mscf/D, and qgc = chart volumetric gas rate, Mscf/D.

Although many gas lift manuals will include gas capacity charts for most typical orifice and choke sizes, numerous charts are unnecessary. The gas capacity for an orifice or choke size can be calculated from a known gas capacity for a given choke size because the calculated volumetric gas throughput rate is directly proportional to the area open to flow for the same gas properties and discharge coefficient. ....................(9)

where

 qg1 = known volumetric gas rate, Mscf/D, d1 = orifice or choke ID for known volumetric gas rate, in., qg2 = unknown volumetric gas rate, Mscf/D, and d2 = orifice or choke ID for unknown volumetric gas rate, in.

If d1 and d2 are fractions, then the denominator of both terms must be the same.

Example Problem 2 Given:

• Injection-gas specific gravity (air = 1.0), γg = 0.7
• Orifice check valve choke size = 1/4-in. ID.
• Injection-gas pressure at valve depth (upstream pressure, P1), PioD = 1,100 psig.
• Flowing-production pressure at valve depth (downstream pressure, P2), PpfD = 900 psig.
• Injection-gas temperature at valve depth (T1), TgD = 140°F.
• Determine the actual volumetric gas throughput of the orifice-check valve:

qgc = 1,200 Mscf/D for 1/4-in.-ID orifice from Fig. 1 (chart value).

Calculate volumetric gas throughput of a 1/2-in.-ID orifice on the basis of the capacity of a 1/4-in.-ID orifice and compare the calculated and chart values (1,200 Mscf/D from Fig. 1 for 1/4-in.-ID orifice),

qgc = 4,800 Mscf/D for 1/2-in.-ID orifice from Fig. 2. There have been misleading references in the literature to the validity of the Thornhill-Craver equation related to gas lift installation design and operation. It is not the equation that is in error. The assumption that a gas lift valve is fully open for all injection-gas throughput calculations is incorrect in most instances. An unloading or operating gas lift valve is seldom fully open. The Thornhill-Craver equation would yield a reasonably accurate injection-gas rate through an operating valve if the actual equivalent port area open to injection-gas flow and the correct discharge coefficient were used in the equation.

## Gas volume stored within a conduit

Typical applications for gas volume calculations are given next.

1. The volume of injection gas required to fill the production conduit and to displace a liquid slug to the surface for intermittent gas lift operations.
2. The volume of injection gas available, or removed, from a casing annulus on the basis of a change in the casing pressure during an intermittent injection-gas cycle (particularly important for design calculations using choke control of the injection gas).
3. The capacity calculations for storage, or retention, of the injection gas in the low- and high-pressure systems in a closed, rotative gas lift system.

The gas capacity and volume calculations are based on an equation of state for real gases. ....................(10)

where

 P = pressure, psia, V = volume or capacity, ft3, z = compressibility factor based on P and T, dimensionless, n = number of pound-moles, lbm mol, R = universal gas constant = , and T = gas temperature, °R.

The volume of gas required to fill a conduit can be calculated with Eq. 11. ....................(11)

where

 Vgsc = volume of gas at standard conditions, scf, Vc = physical capacity of conduit, ft3, = average gas-column pressure, psia, Psc = standard pressure base, psia, = average gas-column temperature, °R, Tsc = standard temperature base, °R, and = compressibility factor based on average pressure, , and average temperature, , dimensionless.

Also, the volume of gas can be calculated by solving for the number of pound-moles in Eq. 10 and by converting the pound-moles to standard cubic feet using Avogadro’s principle which states that 1 lbm-mole of any gas occupies approximately 379 scf at 14.7 psia and 60°F. Average values for pressure and temperature based on surface and bottomhole values and the corresponding compressibility factor must be used in the equation for inclined conduits.

A gas volume equation for pressure difference can be written as ....................(12)

where subscripts 1 and 2 refer to the high and the low average pressure and the corresponding compressibility factor, respectively, and the average gas temperature does not change. If the conduit is horizontal, average pressures and temperature are the surface values in Eqs. 11 and 12. The average temperature of a gas column in the casing is assumed to be the same at the instant a gas lift valve opens or closes. Eq. 12 may be simplified by using one compressibility factor for an average of the average pressures. This assumption is particularly applicable for very little change at high pressure.

Approximate estimations and questionable field data do not warrant detailed calculations. The approximate volume of gas required for a given change in pressure within a conduit can be calculated with Eq. 13. ....................(13)

where

Vgx is the approximate gas volume at standard conditions, scf.

The ratio of the standard to the average temperature, which is less than unity in most cases, tends to offset the reciprocal of the compressibility factor that is greater than unity. This compensation decreases the error from not including several variables in the approximate equation.